Lack of isomorphic embedding from Lorentz function spaces into Lorentz operator ideals

Abstract: In the present paper, we study the existence of isomorphic embeddings from Lorentz
function spaces L_{p_1,q_1} into Lorentz ideals C_{p_2,q_2} in B(H). In particular, we show that, for
p, q ∈ (0, ∞), the quasi-Banach function space L_{p,q} (0, 1) isomorphically embeds into the ideal
C_{p,q} if and only if (p, q) = (2, 2). This extends several existing results in the literature and
answers a question due to Astashkin et. al. [6].